3.1200 \(\int \frac {(d+e x^2)^{5/2} (a+b \tan ^{-1}(c x))}{x^4} \, dx\)

Optimal. Leaf size=114 \[ b \text {Int}\left (\frac {\tan ^{-1}(c x) \left (d+e x^2\right )^{5/2}}{x^4},x\right )+\frac {5}{2} a d e^{3/2} \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )+\frac {5}{2} a e^2 x \sqrt {d+e x^2}-\frac {5 a e \left (d+e x^2\right )^{3/2}}{3 x}-\frac {a \left (d+e x^2\right )^{5/2}}{3 x^3} \]

[Out]

-5/3*a*e*(e*x^2+d)^(3/2)/x-1/3*a*(e*x^2+d)^(5/2)/x^3+5/2*a*d*e^(3/2)*arctanh(x*e^(1/2)/(e*x^2+d)^(1/2))+5/2*a*
e^2*x*(e*x^2+d)^(1/2)+b*Unintegrable((e*x^2+d)^(5/2)*arctan(c*x)/x^4,x)

________________________________________________________________________________________

Rubi [A]  time = 0.19, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\left (d+e x^2\right )^{5/2} \left (a+b \tan ^{-1}(c x)\right )}{x^4} \, dx \]

Verification is Not applicable to the result.

[In]

Int[((d + e*x^2)^(5/2)*(a + b*ArcTan[c*x]))/x^4,x]

[Out]

(5*a*e^2*x*Sqrt[d + e*x^2])/2 - (5*a*e*(d + e*x^2)^(3/2))/(3*x) - (a*(d + e*x^2)^(5/2))/(3*x^3) + (5*a*d*e^(3/
2)*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/2 + b*Defer[Int][((d + e*x^2)^(5/2)*ArcTan[c*x])/x^4, x]

Rubi steps

\begin {align*} \int \frac {\left (d+e x^2\right )^{5/2} \left (a+b \tan ^{-1}(c x)\right )}{x^4} \, dx &=a \int \frac {\left (d+e x^2\right )^{5/2}}{x^4} \, dx+b \int \frac {\left (d+e x^2\right )^{5/2} \tan ^{-1}(c x)}{x^4} \, dx\\ &=-\frac {a \left (d+e x^2\right )^{5/2}}{3 x^3}+b \int \frac {\left (d+e x^2\right )^{5/2} \tan ^{-1}(c x)}{x^4} \, dx+\frac {1}{3} (5 a e) \int \frac {\left (d+e x^2\right )^{3/2}}{x^2} \, dx\\ &=-\frac {5 a e \left (d+e x^2\right )^{3/2}}{3 x}-\frac {a \left (d+e x^2\right )^{5/2}}{3 x^3}+b \int \frac {\left (d+e x^2\right )^{5/2} \tan ^{-1}(c x)}{x^4} \, dx+\left (5 a e^2\right ) \int \sqrt {d+e x^2} \, dx\\ &=\frac {5}{2} a e^2 x \sqrt {d+e x^2}-\frac {5 a e \left (d+e x^2\right )^{3/2}}{3 x}-\frac {a \left (d+e x^2\right )^{5/2}}{3 x^3}+b \int \frac {\left (d+e x^2\right )^{5/2} \tan ^{-1}(c x)}{x^4} \, dx+\frac {1}{2} \left (5 a d e^2\right ) \int \frac {1}{\sqrt {d+e x^2}} \, dx\\ &=\frac {5}{2} a e^2 x \sqrt {d+e x^2}-\frac {5 a e \left (d+e x^2\right )^{3/2}}{3 x}-\frac {a \left (d+e x^2\right )^{5/2}}{3 x^3}+b \int \frac {\left (d+e x^2\right )^{5/2} \tan ^{-1}(c x)}{x^4} \, dx+\frac {1}{2} \left (5 a d e^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-e x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right )\\ &=\frac {5}{2} a e^2 x \sqrt {d+e x^2}-\frac {5 a e \left (d+e x^2\right )^{3/2}}{3 x}-\frac {a \left (d+e x^2\right )^{5/2}}{3 x^3}+\frac {5}{2} a d e^{3/2} \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )+b \int \frac {\left (d+e x^2\right )^{5/2} \tan ^{-1}(c x)}{x^4} \, dx\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 9.55, size = 0, normalized size = 0.00 \[ \int \frac {\left (d+e x^2\right )^{5/2} \left (a+b \tan ^{-1}(c x)\right )}{x^4} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[((d + e*x^2)^(5/2)*(a + b*ArcTan[c*x]))/x^4,x]

[Out]

Integrate[((d + e*x^2)^(5/2)*(a + b*ArcTan[c*x]))/x^4, x]

________________________________________________________________________________________

fricas [A]  time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a e^{2} x^{4} + 2 \, a d e x^{2} + a d^{2} + {\left (b e^{2} x^{4} + 2 \, b d e x^{2} + b d^{2}\right )} \arctan \left (c x\right )\right )} \sqrt {e x^{2} + d}}{x^{4}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^(5/2)*(a+b*arctan(c*x))/x^4,x, algorithm="fricas")

[Out]

integral((a*e^2*x^4 + 2*a*d*e*x^2 + a*d^2 + (b*e^2*x^4 + 2*b*d*e*x^2 + b*d^2)*arctan(c*x))*sqrt(e*x^2 + d)/x^4
, x)

________________________________________________________________________________________

giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^(5/2)*(a+b*arctan(c*x))/x^4,x, algorithm="giac")

[Out]

Timed out

________________________________________________________________________________________

maple [A]  time = 1.08, size = 0, normalized size = 0.00 \[ \int \frac {\left (e \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \arctan \left (c x \right )\right )}{x^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^2+d)^(5/2)*(a+b*arctan(c*x))/x^4,x)

[Out]

int((e*x^2+d)^(5/2)*(a+b*arctan(c*x))/x^4,x)

________________________________________________________________________________________

maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^(5/2)*(a+b*arctan(c*x))/x^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e-c^2*d>0)', see `assume?` for
more details)Is e-c^2*d positive or negative?

________________________________________________________________________________________

mupad [A]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^{5/2}}{x^4} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*atan(c*x))*(d + e*x^2)^(5/2))/x^4,x)

[Out]

int(((a + b*atan(c*x))*(d + e*x^2)^(5/2))/x^4, x)

________________________________________________________________________________________

sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {atan}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{\frac {5}{2}}}{x^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x**2+d)**(5/2)*(a+b*atan(c*x))/x**4,x)

[Out]

Integral((a + b*atan(c*x))*(d + e*x**2)**(5/2)/x**4, x)

________________________________________________________________________________________